The use of system transient (or circuit) simulation techniques to design and analyse complex dynamic systems, incorporating mathematical descriptions of the characteristics of component parts of the systems, has become increasingly widespread. Examples include automotive and aerospace products, and electronic products such as mobile telephones and domestic receivers for satellite TV transmissions. As the typical frequency of operation of such products increases it is becoming increasingly important to include the electro-magnetic (EM) behaviour of complex physical structures, such as electronic packages of passive components, in the transient simulation. Such simulation involves for example accurate frequency-domain or time-domain macro-models of the physical structures. These macro-models are analytic expressions of system transfer functions, and approximate the complex EM behaviour of high-speed multi-port systems at the input and output ports in the frequency or time domains by rational functions (i.e. expressions involving the ratio of two functions).
Rational least-squares approximation is essentially a non-linear problem (a problem which is non-linear in terms of the unknowns), so it is often difficult to estimate the system parameters in a fast and accurate manner. Past proposals have attempted to avoid this difficulty by assuming that knowledge about the poles (denominator of the approximation function) is available. In this case, the non-linear problem reduces to a linear problem since the denominator parameters are assumed to be known. In practice, however, this assumption is often not realistic. Rational linear least-squares approximation techniques are known to suffer from poor numerical conditioning if the frequency range is rather broad, or when the macro-model requires a large number of mathematical poles to provide an adequate representation.
Another possible option is the use of non-linear optimization techniques, such as Newton-Gauss type algorithms. This approach is computationally not always efficient, and the solutions may converge to local minima (leading to suboptimal results), even when Levenberg-Marquardt algorithms are used to extend the region of convergence.
Another proposal was to minimize a Kalman- (or Levi-) linearized cost function which is non-quadratic in the system parameters. This formulation involves setting a weighting factor equal to one for all frequencies. This weighting biases the fitted transfer function, and this often results in poor low-frequency fits, due to an undesired overemphasis of high-frequency errors.
The accuracy of rational least-squares approximation and its numerical conditioning are highly dependent on the structure of the system of equations that is set up as part of the procedure. This procedure includes the choice of “basis functions” from which the numerator and the denominator of the rational expression are assembled. If these basis functions are chosen to be a monomial power series basis (1, s, s2, . . . ) for example, a matrix will result which is a Vandermonde matrix, which is notoriously ill-conditioned. Another proposal is the use as basis functions of polynomials which are orthogonal with respect to a continuous inner product, such as Chebyshev polynomials. The large variation of the Chebyshev polynomials with increase in order makes it possible to downsize the effects of ill-conditioning. A further proposal involves the use of Forsythe polynomials which are orthonormal with respect to a discrete inner product, defined by the normal equations of the estimator. This implies that a different set of basis functions is used for numerator and denominator. It has been shown that the Forsythe polynomial basis is optimal in a sense that there does not exist any other polynomial basis resulting in a better conditioned form of the normal equations.
An iterative macro-modelling technique is described in “Rational Approximation of Frequency Domain Responses by Vector Fitting”, B. Gustavsen and A. Semlyen, IEEE Transactions on Power Delivery, 14 (3), pp. 1052-1061, July 1999. This technique, called Vector Fitting (VF), is basically a reformulation of Sanathanan-Koerner (SK-) iteration (“Transfer Function Synthesis as a Ratio of Two Complex Polynomials”, C. K. Sanathanan and J. Koerner, IEEE Transactions on Automatic Control, AC-8, pp. 56-58, January 1963) using partial fraction basis functions rather than polynomial basis functions. Initially, the poles of partial fractions are prescribed, and they are relocated in successive iterations until a predetermined “cost function” is minimized. The robustness of the method is mainly due to the use of rational basis functions instead of polynomials, which are numerically advantageous if the prescribed poles are properly chosen. This method has been applied in various contexts, such as power systems and microwave engineering. If the prescribed poles are well chosen, the associated system equations are often well-conditioned. For example, the poles may be selected as complex conjugate pairs on a vertical or skew line, close to the imaginary axis of the complex plane. Owing to the iterative behaviour of the SK-iteration, the prescribed poles are relocated until the poles converge in such way that the SK cost function is converged. In general, this happens with relatively few iterations. Vector fitting has been widely applied to many modelling problems in power systems, high-speed interconnection structures, electronic packages and microwave systems.